Well-Posedness and Symmetries of Strongly Coupled Network Equations

Abstract

We consider a diffusion process on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schroedinger equations on a quantum graph are discussed.

25 pages. Corrected typos and minor changes

References in corpus (2)

• Gaussian estimates for a heat equation on a network
• Variational and Semigroup Methods for Waves and Diffusion in Networks

Cited by in corpus (7)

• Parabolic theory of the discrete p-Laplace operator
• Parabolic systems with coupled boundary conditions
• Long-time behavior of stochastically perturbed neuronal networks
• Diffusion processes in thin tubes and their limits on graphs
• A Frucht's theorem for quantum graphs
• Towards a gauge theory for evolution equations on vector-valued spaces
• Laplacians on quantum hypergraphs